# Spearman's rho - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Spearman's rho | Kruskal-Wallis test | Two way ANOVA |
You cannot compare more than 3 methods |
---|---|---|---|

Variable 1 | Independent/grouping variable | Independent/grouping variables | |

One of ordinal level | One categorical with $I$ independent groups ($I \geqslant 2$) | Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | |

Variable 2 | Dependent variable | Dependent variable | |

One of ordinal level | One of ordinal level | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H _{0}: there is no monotonic relationship between the two variables in the population.
| If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
- H
_{0}: the population medians for the $I$ groups are equal
Formulation 1: - H
_{0}: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups
- H
_{0}: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups.
| ANOVA $F$ tests:
- H
_{0}for main and interaction effects together (model): no main effects and interaction effect - H
_{0}for independent variable A: no main effect for A - H
_{0}for independent variable B: no main effect for B - H
_{0}for the interaction term: no interaction effect between A and B
| |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1} two sided: $\rho_s \neq 0$H _{1} right sided: $\rho_s > 0$H _{1} left sided: $\rho_s < 0$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
- H
_{1}: not all of the population medians for the $I$ groups are equal
Formulation 1: - H
_{1}: the poplation scores in some groups are systematically higher or lower than the population scores in other groups
- H
_{1}: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)
| ANOVA $F$ tests:
- H
_{1}for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect - H
_{1}for independent variable A: there is a main effect for A - H
_{1}for independent variable B: there is a main effect for B - H
_{1}for the interaction term: there is an interaction effect between A and B
| |

Assumptions | Assumptions | Assumptions | |

- Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
| - Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another
| - Within each of the $I \times J$ populations, the scores on the dependent variable are normally distributed
- The standard deviation of the scores on the dependent variable is the same in each of the $I \times J$ populations
- For each of the $I \times J$ groups, the sample is an independent and simple random sample from the population defined by that group. That is, within and between groups, observations are independent of one another
- Equal sample sizes for each group make the interpretation of the ANOVA output easier (unequal sample sizes result in overlap in the sum of squares; this is advanced stuff)
| |

Test statistic | Test statistic | Test statistic | |

$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | For main and interaction effects together (model):
- $F = \dfrac{\mbox{mean square model}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square A}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square B}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square interaction}}{\mbox{mean square error}}$
| |

n.a. | n.a. | Pooled standard deviation | |

- | - | $ \begin{aligned} s_p &= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score} - \mbox{its group mean})^2}{N - (I \times J)}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $ | |

Sampling distribution of $t$ if H_{0} were true | Sampling distribution of $H$ if H_{0} were true | Sampling distribution of $F$ if H_{0} were true | |

Approximately the $t$ distribution with $N - 2$ degrees of freedom | For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | For main and interaction effects together (model):
- $F$ distribution with $(I - 1) + (J - 1) + (I - 1) \times (J - 1)$ (df model, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $I - 1$ (df A, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $J - 1$ (df B, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $(I - 1) \times (J - 1)$ (df interaction, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
| |

Significant? | Significant? | Significant? | |

Two sided:
- Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
- Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
- Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
| - Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or
- Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$
| |

n.a. | n.a. | Effect size | |

- | - | *Proportion variance explained $R^2$:* Proportion variance of the dependent variable $y$ explained by the independent variables and the interaction effect together: $$ \begin{align} R^2 &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}} \end{align} $$ $R^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
*Proportion variance explained $\eta^2$:* Proportion variance of the dependent variable $y$ explained by an independent variable or interaction effect: $$ \begin{align} \eta^2_A &= \dfrac{\mbox{sum of squares A}}{\mbox{sum of squares total}}\\ \\ \eta^2_B &= \dfrac{\mbox{sum of squares B}}{\mbox{sum of squares total}}\\ \\ \eta^2_{int} &= \dfrac{\mbox{sum of squares int}}{\mbox{sum of squares total}} \end{align} $$ $\eta^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
*Proportion variance explained $\omega^2$:* Corrects for the positive bias in $\eta^2$ and is equal to: $$ \begin{align} \omega^2_A &= \dfrac{\mbox{sum of squares A} - \mbox{degrees of freedom A} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_B &= \dfrac{\mbox{sum of squares B} - \mbox{degrees of freedom B} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_{int} &= \dfrac{\mbox{sum of squares int} - \mbox{degrees of freedom int} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \end{align} $$ $\omega^2$ is a better estimate of the explained variance in the population than $\eta^2$. Only for balanced designs (equal sample sizes).
*Proportion variance explained $\eta^2_{partial}$:*$$ \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align} $$
| |

n.a. | n.a. | ANOVA table | |

- | - | ||

n.a. | n.a. | Equivalent to | |

- | - | OLS regression with two categorical independent variables and the interaction term, transformed into $(I - 1)$ + $(J - 1)$ + $(I - 1) \times (J - 1)$ code variables. | |

Example context | Example context | Example context | |

Is there a monotonic relationship between physical health and mental health? | Do people from different religions tend to score differently on social economic status? | Is the average mental health score different between people from a low, moderate, and high economic class? And is the average mental health score different between men and women? And is there an interaction effect between economic class and gender? | |

SPSS | SPSS | SPSS | |

Analyze > Correlate > Bivariate...
- Put your two variables in the box below Variables
- Under Correlation Coefficients, select Spearman
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum
- Continue and click OK
| Analyze > General Linear Model > Univariate...
- Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factor(s)
| |

Jamovi | Jamovi | Jamovi | |

Regression > Correlation Matrix
- Put your two variables in the white box at the right
- Under Correlation Coefficients, select Spearman
- Under Hypothesis, select your alternative hypothesis
| ANOVA > One Way ANOVA - Kruskal-Wallis
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
| ANOVA > ANOVA
- Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factors
| |

Practice questions | Practice questions | Practice questions | |